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MAXIMA AND MINIMA OF FUNCTIONS Lesson 2.2

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Definitions Global extrema : If f(c) f(x) for all x in the domain of f, f(c) is the global maximum value of f. If f(c) f(x) for all x in the domain of f, f(c) is the global minimum value of f. Local extrema: If f(c) f(x) for all x in some open interval containing c, f(c) is a local maximum value of f. If f(c) f(x) for all x in some open interval containing c, f(c) is a local minimum value of f.

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Example 1: Approximate the global and local maximum and minimum on each given domain for the function k defined by k(x) = -2x 4 + 3x 3 + 4x 2 – 5x + 5 a. Set of all real numbers: b. -1 x 1 c. x < -2 Y = button

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k(x) = -2x 4 + 3x 3 + 4x 2 – 5x + 5 Set of all real numbers: -no global min, local min. at x.477, k(x) local/global max. at x -.865, k(x) 9.257, - local max. at x 1.513, k(x) 6.502

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k(x) = -2x 4 + 3x 3 + 4x 2 – 5x x 1 -local/global min at x.477, k(x) local min at endpoint x=-1, k(x) = 9 -local/global max. at x -.865, k(x) 9.257, - local max at endpoint, x = 1, k(x) = 5

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k(x) = -2x 4 + 3x 3 + 4x 2 – 5x + 5 x < -2 - there is no minimum since the function decreases without bound on the interval (-, -2). -There is no maximum because k(x) increases as x increases and there is no greatest value of x on this interval.

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Find the extrema of f(t)=2t 4 +4t + 1 Over [0,) - local/global min: t=0, f(t) = 1 - No local/global max. Over (-3,1): -local/global min: t-.787, f(t) No local/global max.

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Homework Page 91 3, 5, 6, 7 10, 12, 14

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